Localization of an unstable solution of a system of three nonlinear ordinary differential equations with a small parameter

In this article we study certain nonlinear autonomous systems of three nonlinear ordinary differential equations (ODEs) with small parameter $\mu$ such that the two variables $(x,y)$ are fast and the other $z$ is slow. Taking the limit as $\mu \to 0$, this becomes the «degenerate system» that is included in the one-parameter family of the two-dimensional subsystems of fast motions with the parameter $z$ from some interval. It is assumed that there is a monotonic function $\boldsymbol \rho(z)$, which in the three-dimensional phase space of a complete dynamical system defines a parametrization of some arc ${\mathcal L}$ of a slow curve consisting of the family of fixed points of the degenerate subsystems. Let ${\mathcal L}$ have the two points of the Andronov—Hopf bifurcation, in which some stable limit cycles arise and disappear in the two-dimensional subsystems. These bifurcation points divide ${\mathcal L}$ into the three arcs: the two arcs are stable and the third arc between them is unstable. For the complete dynamical system we prove the existence of a trajectory which is located as close as possible to the both stable and unstable branches of the slow curve ${\mathcal L}$ as $\mu$ tends to zero and values of $z$ for the given interval.